Scattering of signal in vertical granular chain with impurities

TL;DR

This paper studies how impulses scatter in a vertical granular chain with impurities, revealing power-law reflection rates, asymmetric behaviors, and transition regions in velocity space for both light and heavy impurities.

Contribution

It provides new insights into impulse scattering dynamics in granular chains with impurities, including detailed analysis of reflection rates and transition behaviors for different impurity masses.

Findings

Reflection rates follow power laws with impurity depth.

Asymmetry in reflection rate versus initial perturbation strength.

Transition regions where exponents for signal speed and reflection rate change smoothly.

Abstract

We investigate the scattering process of impulse from impurities in vertical granular chain system under gravity. Most of the studies are focused on the case of light impurities. The reflection rates follow power laws with the impuity depth. There is a valley in the curve of reflection rate versus the initial perturbation strength and the curve is not symmetric. Some results on the scattering process from heavy impurities are also presented. There is a transition region in the initial velocity space. Through this region the exponent for leading peak changes smoothly from its saturant value in small $ v_i$ side to its saturant value in large $v_i$ side; the exponent for signal speed changes smoothly from its saturant value in small $v_i$ side to its saturant value in large $v_i$ side. Both the peak of the exponent for reflection rate and the valley for the reflection rate versus $v_i$ also locate within this transition region.